aiqbench analysis
This research establishes a rigorous link between Schrödinger bridge theory and continuous quantum measurement, demonstrating that bridge potentials serve as direct feedback signals for optimal control. By connecting entropic optimal transport to weak values and Doob potentials, the work provides a new mathematical framework for synthesizing Hamiltonian controls in monitored quantum systems. This advances the theoretical understanding of how to guide quantum state distributions efficiently using measurement-based feedback loops.
Optimal control theory for measured quantum Schr\"odinger bridges
arXiv:2609.03097v1 Announce Type: new Abstract: Schr\"odinger bridges and entropic optimal transport are usually formulated as stochastic interpolation problems between initial and final probability distributions. In their computational form, the bridge potentials are obtained by Sinkhorn or iterative proportional fitting, and are often regarded as auxiliary scaling functions. We show that for continuously monitored quantum systems, these potentials acquire a direct measurement-theoretic meaning. The mathematical structure of conditional quantum trajectory theory induces a Fokker--Planck equation on quantum state space. Conditioning this diffusion on a terminal distribution or a terminal measurement effect produces a Doob/Sinkhorn potential whose directional derivative along a unitary control vector field is the imaginary part of a generalized weak value. The same construction connects the Schr\"odinger-bridge viewpoint to the optimal-path framework for continuously monitored trajectories: the backward bridge potential plays the role of an effect-like costate, and its weak-value directional derivative gives the local control signal. By specifying the desired endpoint distribution, the induced drift produced by the Schr\"odinger bridge solution is the control solution that minimizes the quadratic cost feedback law, guiding the distribution to its desired endpoint. We quantify the control score of the available control Hamiltonian as a logarithmic directional derivative of the bridge potential, or an imaginary weak value. Three explicit examples identify weak measurement as a natural entropic regularization mechanism for quantum state transport and gives a route from Sinkhorn scaling to quantum feedback and Hamiltonian control synthesis for practical optimal control.
Abstract:Schrödinger bridges and entropic optimal transport are usually formulated as stochastic interpolation problems between initial and final probability distributions. In their computational form, the bridge potentials are obtained by Sinkhorn or iterative proportional fitting, and are often regarded as auxiliary scaling functions. We show that for continuously monitored quantum systems, these potentials acquire a direct measurement-theoretic meaning. The mathematical structure of conditional quantum trajectory theory induces a Fokker--Planck equation on quantum state space. Conditioning this diffusion on a terminal distribution or a terminal measurement effect produces a Doob/Sinkhorn potential whose directional derivative along a unitary control vector field is the imaginary part of a generalized weak value. The same construction connects the Schrödinger-bridge viewpoint to the optimal-path framework for continuously monitored trajectories: the backward bridge potential plays the role of an effect-like costate, and its weak-value directional derivative gives the local control signal. By specifying the desired endpoint distribution, the induced drift produced by the Schrödinger bridge solution is the control solution that minimizes the quadratic cost feedback law, guiding the distribution to its desired endpoint. We quantify the control score of the available control Hamiltonian as a logarithmic directional derivative of the bridge potential, or an imaginary weak value. Three explicit examples identify weak measurement as a natural entropic regularization mechanism for quantum state transport and gives a route from Sinkhorn scaling to quantum feedback and Hamiltonian control synthesis for practical optimal control.
Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech)
arXiv:2609.03097 [quant-ph]
(or arXiv:2609.03097v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2609.03097 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 331 words
